Configurations and invariant nets for amenable hypergroups and related algebras
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Abstract:
Let $H$ be a hypergroup with left Haar measure. The amenability of $H$ can be characterized by the existence of nets of positive, norm one functions in $L^1(H)$ which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of $H$ using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in $L^1(H)$ which tend to left invariance weakly, but not in norm.
We define the semidirect product of $H$ with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product.
These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra.
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Additional Information
- Benjamin Willson
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- Address at time of publication: Department of Mathematics, Hanyang University, Seoul, South Korea
- Email: bwillson@ualberta.ca
- Received by editor(s): June 1, 2011
- Received by editor(s) in revised form: October 22, 2011
- Published electronically: July 16, 2014
- Additional Notes: The financial assistance of NSERC in the form of a Postgraduate Scholarship(Doctoral) and funding through a Discovery Grant awarded to Professor Anthony To-Ming Lau is gratefully acknowledged.
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 5087-5112
- MSC (2010): Primary 43A62; Secondary 43A07, 20N20
- DOI: https://doi.org/10.1090/S0002-9947-2014-05731-0
- MathSciNet review: 3240918